uspgr2wlkeq - Intuitionistic Logic Explorer

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Theorem uspgr2wlkeq 16486

Description: Conditions for two walks within the same simple pseudograph being the same. It is sufficient that the vertices (in the same order) are identical. (Contributed by AV, 3-Jul-2018.) (Revised by AV, 14-Apr-2021.)

**Assertion**
Ref	Expression
uspgr2wlkeq	⊢ ((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 𝑁 = (♯‘(1^st ‘𝐴))) → (𝐴 = 𝐵 ↔ (𝑁 = (♯‘(1^st ‘𝐵)) ∧ ∀𝑦 ∈ (0...𝑁)((2^nd ‘𝐴)‘𝑦) = ((2^nd ‘𝐵)‘𝑦))))

Distinct variable groups: 𝑦,𝐴 𝑦,𝐵 𝑦,𝐺 𝑦,𝑁

Proof of Theorem uspgr2wlkeq Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		3anan32 1016	. . 3 ⊢ ((𝑁 = (♯‘(1^st ‘𝐵)) ∧ ∀𝑦 ∈ (0..^𝑁)((1^st ‘𝐴)‘𝑦) = ((1^st ‘𝐵)‘𝑦) ∧ ∀𝑦 ∈ (0...𝑁)((2^nd ‘𝐴)‘𝑦) = ((2^nd ‘𝐵)‘𝑦)) ↔ ((𝑁 = (♯‘(1^st ‘𝐵)) ∧ ∀𝑦 ∈ (0...𝑁)((2^nd ‘𝐴)‘𝑦) = ((2^nd ‘𝐵)‘𝑦)) ∧ ∀𝑦 ∈ (0..^𝑁)((1^st ‘𝐴)‘𝑦) = ((1^st ‘𝐵)‘𝑦)))
2	1	a1i 9	. 2 ⊢ ((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 𝑁 = (♯‘(1^st ‘𝐴))) → ((𝑁 = (♯‘(1^st ‘𝐵)) ∧ ∀𝑦 ∈ (0..^𝑁)((1^st ‘𝐴)‘𝑦) = ((1^st ‘𝐵)‘𝑦) ∧ ∀𝑦 ∈ (0...𝑁)((2^nd ‘𝐴)‘𝑦) = ((2^nd ‘𝐵)‘𝑦)) ↔ ((𝑁 = (♯‘(1^st ‘𝐵)) ∧ ∀𝑦 ∈ (0...𝑁)((2^nd ‘𝐴)‘𝑦) = ((2^nd ‘𝐵)‘𝑦)) ∧ ∀𝑦 ∈ (0..^𝑁)((1^st ‘𝐴)‘𝑦) = ((1^st ‘𝐵)‘𝑦))))
3		wlkeq 16475	. . . 4 ⊢ ((𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺) ∧ 𝑁 = (♯‘(1^st ‘𝐴))) → (𝐴 = 𝐵 ↔ (𝑁 = (♯‘(1^st ‘𝐵)) ∧ ∀𝑦 ∈ (0..^𝑁)((1^st ‘𝐴)‘𝑦) = ((1^st ‘𝐵)‘𝑦) ∧ ∀𝑦 ∈ (0...𝑁)((2^nd ‘𝐴)‘𝑦) = ((2^nd ‘𝐵)‘𝑦))))
4	3	3expa 1230	. . 3 ⊢ (((𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 𝑁 = (♯‘(1^st ‘𝐴))) → (𝐴 = 𝐵 ↔ (𝑁 = (♯‘(1^st ‘𝐵)) ∧ ∀𝑦 ∈ (0..^𝑁)((1^st ‘𝐴)‘𝑦) = ((1^st ‘𝐵)‘𝑦) ∧ ∀𝑦 ∈ (0...𝑁)((2^nd ‘𝐴)‘𝑦) = ((2^nd ‘𝐵)‘𝑦))))
5	4	3adant1 1042	. 2 ⊢ ((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 𝑁 = (♯‘(1^st ‘𝐴))) → (𝐴 = 𝐵 ↔ (𝑁 = (♯‘(1^st ‘𝐵)) ∧ ∀𝑦 ∈ (0..^𝑁)((1^st ‘𝐴)‘𝑦) = ((1^st ‘𝐵)‘𝑦) ∧ ∀𝑦 ∈ (0...𝑁)((2^nd ‘𝐴)‘𝑦) = ((2^nd ‘𝐵)‘𝑦))))
6		fzofzp1 10594	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (0..^𝑁) → (𝑥 + 1) ∈ (0...𝑁))
7	6	adantl 277	. . . . . . . . . . 11 ⊢ ((((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 𝑁 = (♯‘(1^st ‘𝐴))) ∧ 𝑁 = (♯‘(1^st ‘𝐵))) ∧ 𝑥 ∈ (0..^𝑁)) → (𝑥 + 1) ∈ (0...𝑁))
8		fveq2 5675	. . . . . . . . . . . . 13 ⊢ (𝑦 = (𝑥 + 1) → ((2^nd ‘𝐴)‘𝑦) = ((2^nd ‘𝐴)‘(𝑥 + 1)))
9		fveq2 5675	. . . . . . . . . . . . 13 ⊢ (𝑦 = (𝑥 + 1) → ((2^nd ‘𝐵)‘𝑦) = ((2^nd ‘𝐵)‘(𝑥 + 1)))
10	8, 9	eqeq12d 2249	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝑥 + 1) → (((2^nd ‘𝐴)‘𝑦) = ((2^nd ‘𝐵)‘𝑦) ↔ ((2^nd ‘𝐴)‘(𝑥 + 1)) = ((2^nd ‘𝐵)‘(𝑥 + 1))))
11	10	adantl 277	. . . . . . . . . . 11 ⊢ (((((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 𝑁 = (♯‘(1^st ‘𝐴))) ∧ 𝑁 = (♯‘(1^st ‘𝐵))) ∧ 𝑥 ∈ (0..^𝑁)) ∧ 𝑦 = (𝑥 + 1)) → (((2^nd ‘𝐴)‘𝑦) = ((2^nd ‘𝐵)‘𝑦) ↔ ((2^nd ‘𝐴)‘(𝑥 + 1)) = ((2^nd ‘𝐵)‘(𝑥 + 1))))
12	7, 11	rspcdv 2926	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 𝑁 = (♯‘(1^st ‘𝐴))) ∧ 𝑁 = (♯‘(1^st ‘𝐵))) ∧ 𝑥 ∈ (0..^𝑁)) → (∀𝑦 ∈ (0...𝑁)((2^nd ‘𝐴)‘𝑦) = ((2^nd ‘𝐵)‘𝑦) → ((2^nd ‘𝐴)‘(𝑥 + 1)) = ((2^nd ‘𝐵)‘(𝑥 + 1))))
13	12	impancom 260	. . . . . . . . 9 ⊢ ((((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 𝑁 = (♯‘(1^st ‘𝐴))) ∧ 𝑁 = (♯‘(1^st ‘𝐵))) ∧ ∀𝑦 ∈ (0...𝑁)((2^nd ‘𝐴)‘𝑦) = ((2^nd ‘𝐵)‘𝑦)) → (𝑥 ∈ (0..^𝑁) → ((2^nd ‘𝐴)‘(𝑥 + 1)) = ((2^nd ‘𝐵)‘(𝑥 + 1))))
14	13	ralrimiv 2616	. . . . . . . 8 ⊢ ((((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 𝑁 = (♯‘(1^st ‘𝐴))) ∧ 𝑁 = (♯‘(1^st ‘𝐵))) ∧ ∀𝑦 ∈ (0...𝑁)((2^nd ‘𝐴)‘𝑦) = ((2^nd ‘𝐵)‘𝑦)) → ∀𝑥 ∈ (0..^𝑁)((2^nd ‘𝐴)‘(𝑥 + 1)) = ((2^nd ‘𝐵)‘(𝑥 + 1)))
15		fvoveq1 6081	. . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → ((2^nd ‘𝐴)‘(𝑦 + 1)) = ((2^nd ‘𝐴)‘(𝑥 + 1)))
16		fvoveq1 6081	. . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → ((2^nd ‘𝐵)‘(𝑦 + 1)) = ((2^nd ‘𝐵)‘(𝑥 + 1)))
17	15, 16	eqeq12d 2249	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → (((2^nd ‘𝐴)‘(𝑦 + 1)) = ((2^nd ‘𝐵)‘(𝑦 + 1)) ↔ ((2^nd ‘𝐴)‘(𝑥 + 1)) = ((2^nd ‘𝐵)‘(𝑥 + 1))))
18	17	cbvralvw 2784	. . . . . . . 8 ⊢ (∀𝑦 ∈ (0..^𝑁)((2^nd ‘𝐴)‘(𝑦 + 1)) = ((2^nd ‘𝐵)‘(𝑦 + 1)) ↔ ∀𝑥 ∈ (0..^𝑁)((2^nd ‘𝐴)‘(𝑥 + 1)) = ((2^nd ‘𝐵)‘(𝑥 + 1)))
19	14, 18	sylibr 134	. . . . . . 7 ⊢ ((((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 𝑁 = (♯‘(1^st ‘𝐴))) ∧ 𝑁 = (♯‘(1^st ‘𝐵))) ∧ ∀𝑦 ∈ (0...𝑁)((2^nd ‘𝐴)‘𝑦) = ((2^nd ‘𝐵)‘𝑦)) → ∀𝑦 ∈ (0..^𝑁)((2^nd ‘𝐴)‘(𝑦 + 1)) = ((2^nd ‘𝐵)‘(𝑦 + 1)))
20		fzossfz 10522	. . . . . . . . . 10 ⊢ (0..^𝑁) ⊆ (0...𝑁)
21		ssralv 3306	. . . . . . . . . 10 ⊢ ((0..^𝑁) ⊆ (0...𝑁) → (∀𝑦 ∈ (0...𝑁)((2^nd ‘𝐴)‘𝑦) = ((2^nd ‘𝐵)‘𝑦) → ∀𝑦 ∈ (0..^𝑁)((2^nd ‘𝐴)‘𝑦) = ((2^nd ‘𝐵)‘𝑦)))
22	20, 21	mp1i 10	. . . . . . . . 9 ⊢ (((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 𝑁 = (♯‘(1^st ‘𝐴))) ∧ 𝑁 = (♯‘(1^st ‘𝐵))) → (∀𝑦 ∈ (0...𝑁)((2^nd ‘𝐴)‘𝑦) = ((2^nd ‘𝐵)‘𝑦) → ∀𝑦 ∈ (0..^𝑁)((2^nd ‘𝐴)‘𝑦) = ((2^nd ‘𝐵)‘𝑦)))
23		r19.26 2671	. . . . . . . . . . 11 ⊢ (∀𝑦 ∈ (0..^𝑁)(((2^nd ‘𝐴)‘𝑦) = ((2^nd ‘𝐵)‘𝑦) ∧ ((2^nd ‘𝐴)‘(𝑦 + 1)) = ((2^nd ‘𝐵)‘(𝑦 + 1))) ↔ (∀𝑦 ∈ (0..^𝑁)((2^nd ‘𝐴)‘𝑦) = ((2^nd ‘𝐵)‘𝑦) ∧ ∀𝑦 ∈ (0..^𝑁)((2^nd ‘𝐴)‘(𝑦 + 1)) = ((2^nd ‘𝐵)‘(𝑦 + 1))))
24		preq12 3775	. . . . . . . . . . . . 13 ⊢ ((((2^nd ‘𝐴)‘𝑦) = ((2^nd ‘𝐵)‘𝑦) ∧ ((2^nd ‘𝐴)‘(𝑦 + 1)) = ((2^nd ‘𝐵)‘(𝑦 + 1))) → {((2^nd ‘𝐴)‘𝑦), ((2^nd ‘𝐴)‘(𝑦 + 1))} = {((2^nd ‘𝐵)‘𝑦), ((2^nd ‘𝐵)‘(𝑦 + 1))})
25	24	a1i 9	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 𝑁 = (♯‘(1^st ‘𝐴))) ∧ 𝑁 = (♯‘(1^st ‘𝐵))) → ((((2^nd ‘𝐴)‘𝑦) = ((2^nd ‘𝐵)‘𝑦) ∧ ((2^nd ‘𝐴)‘(𝑦 + 1)) = ((2^nd ‘𝐵)‘(𝑦 + 1))) → {((2^nd ‘𝐴)‘𝑦), ((2^nd ‘𝐴)‘(𝑦 + 1))} = {((2^nd ‘𝐵)‘𝑦), ((2^nd ‘𝐵)‘(𝑦 + 1))}))
26	25	ralimdv 2612	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 𝑁 = (♯‘(1^st ‘𝐴))) ∧ 𝑁 = (♯‘(1^st ‘𝐵))) → (∀𝑦 ∈ (0..^𝑁)(((2^nd ‘𝐴)‘𝑦) = ((2^nd ‘𝐵)‘𝑦) ∧ ((2^nd ‘𝐴)‘(𝑦 + 1)) = ((2^nd ‘𝐵)‘(𝑦 + 1))) → ∀𝑦 ∈ (0..^𝑁){((2^nd ‘𝐴)‘𝑦), ((2^nd ‘𝐴)‘(𝑦 + 1))} = {((2^nd ‘𝐵)‘𝑦), ((2^nd ‘𝐵)‘(𝑦 + 1))}))
27	23, 26	biimtrrid 153	. . . . . . . . . 10 ⊢ (((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 𝑁 = (♯‘(1^st ‘𝐴))) ∧ 𝑁 = (♯‘(1^st ‘𝐵))) → ((∀𝑦 ∈ (0..^𝑁)((2^nd ‘𝐴)‘𝑦) = ((2^nd ‘𝐵)‘𝑦) ∧ ∀𝑦 ∈ (0..^𝑁)((2^nd ‘𝐴)‘(𝑦 + 1)) = ((2^nd ‘𝐵)‘(𝑦 + 1))) → ∀𝑦 ∈ (0..^𝑁){((2^nd ‘𝐴)‘𝑦), ((2^nd ‘𝐴)‘(𝑦 + 1))} = {((2^nd ‘𝐵)‘𝑦), ((2^nd ‘𝐵)‘(𝑦 + 1))}))
28	27	expd 258	. . . . . . . . 9 ⊢ (((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 𝑁 = (♯‘(1^st ‘𝐴))) ∧ 𝑁 = (♯‘(1^st ‘𝐵))) → (∀𝑦 ∈ (0..^𝑁)((2^nd ‘𝐴)‘𝑦) = ((2^nd ‘𝐵)‘𝑦) → (∀𝑦 ∈ (0..^𝑁)((2^nd ‘𝐴)‘(𝑦 + 1)) = ((2^nd ‘𝐵)‘(𝑦 + 1)) → ∀𝑦 ∈ (0..^𝑁){((2^nd ‘𝐴)‘𝑦), ((2^nd ‘𝐴)‘(𝑦 + 1))} = {((2^nd ‘𝐵)‘𝑦), ((2^nd ‘𝐵)‘(𝑦 + 1))})))
29	22, 28	syld 45	. . . . . . . 8 ⊢ (((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 𝑁 = (♯‘(1^st ‘𝐴))) ∧ 𝑁 = (♯‘(1^st ‘𝐵))) → (∀𝑦 ∈ (0...𝑁)((2^nd ‘𝐴)‘𝑦) = ((2^nd ‘𝐵)‘𝑦) → (∀𝑦 ∈ (0..^𝑁)((2^nd ‘𝐴)‘(𝑦 + 1)) = ((2^nd ‘𝐵)‘(𝑦 + 1)) → ∀𝑦 ∈ (0..^𝑁){((2^nd ‘𝐴)‘𝑦), ((2^nd ‘𝐴)‘(𝑦 + 1))} = {((2^nd ‘𝐵)‘𝑦), ((2^nd ‘𝐵)‘(𝑦 + 1))})))
30	29	imp 124	. . . . . . 7 ⊢ ((((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 𝑁 = (♯‘(1^st ‘𝐴))) ∧ 𝑁 = (♯‘(1^st ‘𝐵))) ∧ ∀𝑦 ∈ (0...𝑁)((2^nd ‘𝐴)‘𝑦) = ((2^nd ‘𝐵)‘𝑦)) → (∀𝑦 ∈ (0..^𝑁)((2^nd ‘𝐴)‘(𝑦 + 1)) = ((2^nd ‘𝐵)‘(𝑦 + 1)) → ∀𝑦 ∈ (0..^𝑁){((2^nd ‘𝐴)‘𝑦), ((2^nd ‘𝐴)‘(𝑦 + 1))} = {((2^nd ‘𝐵)‘𝑦), ((2^nd ‘𝐵)‘(𝑦 + 1))}))
31	19, 30	mpd 13	. . . . . 6 ⊢ ((((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 𝑁 = (♯‘(1^st ‘𝐴))) ∧ 𝑁 = (♯‘(1^st ‘𝐵))) ∧ ∀𝑦 ∈ (0...𝑁)((2^nd ‘𝐴)‘𝑦) = ((2^nd ‘𝐵)‘𝑦)) → ∀𝑦 ∈ (0..^𝑁){((2^nd ‘𝐴)‘𝑦), ((2^nd ‘𝐴)‘(𝑦 + 1))} = {((2^nd ‘𝐵)‘𝑦), ((2^nd ‘𝐵)‘(𝑦 + 1))})
32	31	ex 115	. . . . 5 ⊢ (((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 𝑁 = (♯‘(1^st ‘𝐴))) ∧ 𝑁 = (♯‘(1^st ‘𝐵))) → (∀𝑦 ∈ (0...𝑁)((2^nd ‘𝐴)‘𝑦) = ((2^nd ‘𝐵)‘𝑦) → ∀𝑦 ∈ (0..^𝑁){((2^nd ‘𝐴)‘𝑦), ((2^nd ‘𝐴)‘(𝑦 + 1))} = {((2^nd ‘𝐵)‘𝑦), ((2^nd ‘𝐵)‘(𝑦 + 1))}))
33		uspgrupgr 16302	. . . . . . . 8 ⊢ (𝐺 ∈ USPGraph → 𝐺 ∈ UPGraph)
34		eqid 2234	. . . . . . . . . 10 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
35		eqid 2234	. . . . . . . . . 10 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
36		eqid 2234	. . . . . . . . . 10 ⊢ (1^st ‘𝐴) = (1^st ‘𝐴)
37		eqid 2234	. . . . . . . . . 10 ⊢ (2^nd ‘𝐴) = (2^nd ‘𝐴)
38	34, 35, 36, 37	upgrwlkcompim 16483	. . . . . . . . 9 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐴 ∈ (Walks‘𝐺)) → ((1^st ‘𝐴) ∈ Word dom (iEdg‘𝐺) ∧ (2^nd ‘𝐴):(0...(♯‘(1^st ‘𝐴)))⟶(Vtx‘𝐺) ∧ ∀𝑦 ∈ (0..^(♯‘(1^st ‘𝐴)))((iEdg‘𝐺)‘((1^st ‘𝐴)‘𝑦)) = {((2^nd ‘𝐴)‘𝑦), ((2^nd ‘𝐴)‘(𝑦 + 1))}))
39	38	ex 115	. . . . . . . 8 ⊢ (𝐺 ∈ UPGraph → (𝐴 ∈ (Walks‘𝐺) → ((1^st ‘𝐴) ∈ Word dom (iEdg‘𝐺) ∧ (2^nd ‘𝐴):(0...(♯‘(1^st ‘𝐴)))⟶(Vtx‘𝐺) ∧ ∀𝑦 ∈ (0..^(♯‘(1^st ‘𝐴)))((iEdg‘𝐺)‘((1^st ‘𝐴)‘𝑦)) = {((2^nd ‘𝐴)‘𝑦), ((2^nd ‘𝐴)‘(𝑦 + 1))})))
40	33, 39	syl 14	. . . . . . 7 ⊢ (𝐺 ∈ USPGraph → (𝐴 ∈ (Walks‘𝐺) → ((1^st ‘𝐴) ∈ Word dom (iEdg‘𝐺) ∧ (2^nd ‘𝐴):(0...(♯‘(1^st ‘𝐴)))⟶(Vtx‘𝐺) ∧ ∀𝑦 ∈ (0..^(♯‘(1^st ‘𝐴)))((iEdg‘𝐺)‘((1^st ‘𝐴)‘𝑦)) = {((2^nd ‘𝐴)‘𝑦), ((2^nd ‘𝐴)‘(𝑦 + 1))})))
41		eqid 2234	. . . . . . . . . 10 ⊢ (1^st ‘𝐵) = (1^st ‘𝐵)
42		eqid 2234	. . . . . . . . . 10 ⊢ (2^nd ‘𝐵) = (2^nd ‘𝐵)
43	34, 35, 41, 42	upgrwlkcompim 16483	. . . . . . . . 9 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐵 ∈ (Walks‘𝐺)) → ((1^st ‘𝐵) ∈ Word dom (iEdg‘𝐺) ∧ (2^nd ‘𝐵):(0...(♯‘(1^st ‘𝐵)))⟶(Vtx‘𝐺) ∧ ∀𝑦 ∈ (0..^(♯‘(1^st ‘𝐵)))((iEdg‘𝐺)‘((1^st ‘𝐵)‘𝑦)) = {((2^nd ‘𝐵)‘𝑦), ((2^nd ‘𝐵)‘(𝑦 + 1))}))
44	43	ex 115	. . . . . . . 8 ⊢ (𝐺 ∈ UPGraph → (𝐵 ∈ (Walks‘𝐺) → ((1^st ‘𝐵) ∈ Word dom (iEdg‘𝐺) ∧ (2^nd ‘𝐵):(0...(♯‘(1^st ‘𝐵)))⟶(Vtx‘𝐺) ∧ ∀𝑦 ∈ (0..^(♯‘(1^st ‘𝐵)))((iEdg‘𝐺)‘((1^st ‘𝐵)‘𝑦)) = {((2^nd ‘𝐵)‘𝑦), ((2^nd ‘𝐵)‘(𝑦 + 1))})))
45	33, 44	syl 14	. . . . . . 7 ⊢ (𝐺 ∈ USPGraph → (𝐵 ∈ (Walks‘𝐺) → ((1^st ‘𝐵) ∈ Word dom (iEdg‘𝐺) ∧ (2^nd ‘𝐵):(0...(♯‘(1^st ‘𝐵)))⟶(Vtx‘𝐺) ∧ ∀𝑦 ∈ (0..^(♯‘(1^st ‘𝐵)))((iEdg‘𝐺)‘((1^st ‘𝐵)‘𝑦)) = {((2^nd ‘𝐵)‘𝑦), ((2^nd ‘𝐵)‘(𝑦 + 1))})))
46		oveq2 6066	. . . . . . . . . . . . . . . . . . 19 ⊢ ((♯‘(1^st ‘𝐵)) = 𝑁 → (0..^(♯‘(1^st ‘𝐵))) = (0..^𝑁))
47	46	eqcoms 2237	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 = (♯‘(1^st ‘𝐵)) → (0..^(♯‘(1^st ‘𝐵))) = (0..^𝑁))
48	47	raleqdv 2749	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 = (♯‘(1^st ‘𝐵)) → (∀𝑦 ∈ (0..^(♯‘(1^st ‘𝐵)))((iEdg‘𝐺)‘((1^st ‘𝐵)‘𝑦)) = {((2^nd ‘𝐵)‘𝑦), ((2^nd ‘𝐵)‘(𝑦 + 1))} ↔ ∀𝑦 ∈ (0..^𝑁)((iEdg‘𝐺)‘((1^st ‘𝐵)‘𝑦)) = {((2^nd ‘𝐵)‘𝑦), ((2^nd ‘𝐵)‘(𝑦 + 1))}))
49		oveq2 6066	. . . . . . . . . . . . . . . . . . 19 ⊢ ((♯‘(1^st ‘𝐴)) = 𝑁 → (0..^(♯‘(1^st ‘𝐴))) = (0..^𝑁))
50	49	eqcoms 2237	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 = (♯‘(1^st ‘𝐴)) → (0..^(♯‘(1^st ‘𝐴))) = (0..^𝑁))
51	50	raleqdv 2749	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 = (♯‘(1^st ‘𝐴)) → (∀𝑦 ∈ (0..^(♯‘(1^st ‘𝐴)))((iEdg‘𝐺)‘((1^st ‘𝐴)‘𝑦)) = {((2^nd ‘𝐴)‘𝑦), ((2^nd ‘𝐴)‘(𝑦 + 1))} ↔ ∀𝑦 ∈ (0..^𝑁)((iEdg‘𝐺)‘((1^st ‘𝐴)‘𝑦)) = {((2^nd ‘𝐴)‘𝑦), ((2^nd ‘𝐴)‘(𝑦 + 1))}))
52	48, 51	bi2anan9r 611	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 = (♯‘(1^st ‘𝐴)) ∧ 𝑁 = (♯‘(1^st ‘𝐵))) → ((∀𝑦 ∈ (0..^(♯‘(1^st ‘𝐵)))((iEdg‘𝐺)‘((1^st ‘𝐵)‘𝑦)) = {((2^nd ‘𝐵)‘𝑦), ((2^nd ‘𝐵)‘(𝑦 + 1))} ∧ ∀𝑦 ∈ (0..^(♯‘(1^st ‘𝐴)))((iEdg‘𝐺)‘((1^st ‘𝐴)‘𝑦)) = {((2^nd ‘𝐴)‘𝑦), ((2^nd ‘𝐴)‘(𝑦 + 1))}) ↔ (∀𝑦 ∈ (0..^𝑁)((iEdg‘𝐺)‘((1^st ‘𝐵)‘𝑦)) = {((2^nd ‘𝐵)‘𝑦), ((2^nd ‘𝐵)‘(𝑦 + 1))} ∧ ∀𝑦 ∈ (0..^𝑁)((iEdg‘𝐺)‘((1^st ‘𝐴)‘𝑦)) = {((2^nd ‘𝐴)‘𝑦), ((2^nd ‘𝐴)‘(𝑦 + 1))})))
53		r19.26 2671	. . . . . . . . . . . . . . . . 17 ⊢ (∀𝑦 ∈ (0..^𝑁)(((iEdg‘𝐺)‘((1^st ‘𝐵)‘𝑦)) = {((2^nd ‘𝐵)‘𝑦), ((2^nd ‘𝐵)‘(𝑦 + 1))} ∧ ((iEdg‘𝐺)‘((1^st ‘𝐴)‘𝑦)) = {((2^nd ‘𝐴)‘𝑦), ((2^nd ‘𝐴)‘(𝑦 + 1))}) ↔ (∀𝑦 ∈ (0..^𝑁)((iEdg‘𝐺)‘((1^st ‘𝐵)‘𝑦)) = {((2^nd ‘𝐵)‘𝑦), ((2^nd ‘𝐵)‘(𝑦 + 1))} ∧ ∀𝑦 ∈ (0..^𝑁)((iEdg‘𝐺)‘((1^st ‘𝐴)‘𝑦)) = {((2^nd ‘𝐴)‘𝑦), ((2^nd ‘𝐴)‘(𝑦 + 1))}))
54		eqeq2 2244	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ({((2^nd ‘𝐴)‘𝑦), ((2^nd ‘𝐴)‘(𝑦 + 1))} = {((2^nd ‘𝐵)‘𝑦), ((2^nd ‘𝐵)‘(𝑦 + 1))} → (((iEdg‘𝐺)‘((1^st ‘𝐴)‘𝑦)) = {((2^nd ‘𝐴)‘𝑦), ((2^nd ‘𝐴)‘(𝑦 + 1))} ↔ ((iEdg‘𝐺)‘((1^st ‘𝐴)‘𝑦)) = {((2^nd ‘𝐵)‘𝑦), ((2^nd ‘𝐵)‘(𝑦 + 1))}))
55		eqeq2 2244	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ({((2^nd ‘𝐵)‘𝑦), ((2^nd ‘𝐵)‘(𝑦 + 1))} = ((iEdg‘𝐺)‘((1^st ‘𝐴)‘𝑦)) → (((iEdg‘𝐺)‘((1^st ‘𝐵)‘𝑦)) = {((2^nd ‘𝐵)‘𝑦), ((2^nd ‘𝐵)‘(𝑦 + 1))} ↔ ((iEdg‘𝐺)‘((1^st ‘𝐵)‘𝑦)) = ((iEdg‘𝐺)‘((1^st ‘𝐴)‘𝑦))))
56	55	eqcoms 2237	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((iEdg‘𝐺)‘((1^st ‘𝐴)‘𝑦)) = {((2^nd ‘𝐵)‘𝑦), ((2^nd ‘𝐵)‘(𝑦 + 1))} → (((iEdg‘𝐺)‘((1^st ‘𝐵)‘𝑦)) = {((2^nd ‘𝐵)‘𝑦), ((2^nd ‘𝐵)‘(𝑦 + 1))} ↔ ((iEdg‘𝐺)‘((1^st ‘𝐵)‘𝑦)) = ((iEdg‘𝐺)‘((1^st ‘𝐴)‘𝑦))))
57	56	biimpd 144	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((iEdg‘𝐺)‘((1^st ‘𝐴)‘𝑦)) = {((2^nd ‘𝐵)‘𝑦), ((2^nd ‘𝐵)‘(𝑦 + 1))} → (((iEdg‘𝐺)‘((1^st ‘𝐵)‘𝑦)) = {((2^nd ‘𝐵)‘𝑦), ((2^nd ‘𝐵)‘(𝑦 + 1))} → ((iEdg‘𝐺)‘((1^st ‘𝐵)‘𝑦)) = ((iEdg‘𝐺)‘((1^st ‘𝐴)‘𝑦))))
58	54, 57	biimtrdi 163	. . . . . . . . . . . . . . . . . . . 20 ⊢ ({((2^nd ‘𝐴)‘𝑦), ((2^nd ‘𝐴)‘(𝑦 + 1))} = {((2^nd ‘𝐵)‘𝑦), ((2^nd ‘𝐵)‘(𝑦 + 1))} → (((iEdg‘𝐺)‘((1^st ‘𝐴)‘𝑦)) = {((2^nd ‘𝐴)‘𝑦), ((2^nd ‘𝐴)‘(𝑦 + 1))} → (((iEdg‘𝐺)‘((1^st ‘𝐵)‘𝑦)) = {((2^nd ‘𝐵)‘𝑦), ((2^nd ‘𝐵)‘(𝑦 + 1))} → ((iEdg‘𝐺)‘((1^st ‘𝐵)‘𝑦)) = ((iEdg‘𝐺)‘((1^st ‘𝐴)‘𝑦)))))
59	58	com13 80	. . . . . . . . . . . . . . . . . . 19 ⊢ (((iEdg‘𝐺)‘((1^st ‘𝐵)‘𝑦)) = {((2^nd ‘𝐵)‘𝑦), ((2^nd ‘𝐵)‘(𝑦 + 1))} → (((iEdg‘𝐺)‘((1^st ‘𝐴)‘𝑦)) = {((2^nd ‘𝐴)‘𝑦), ((2^nd ‘𝐴)‘(𝑦 + 1))} → ({((2^nd ‘𝐴)‘𝑦), ((2^nd ‘𝐴)‘(𝑦 + 1))} = {((2^nd ‘𝐵)‘𝑦), ((2^nd ‘𝐵)‘(𝑦 + 1))} → ((iEdg‘𝐺)‘((1^st ‘𝐵)‘𝑦)) = ((iEdg‘𝐺)‘((1^st ‘𝐴)‘𝑦)))))
60	59	imp 124	. . . . . . . . . . . . . . . . . 18 ⊢ ((((iEdg‘𝐺)‘((1^st ‘𝐵)‘𝑦)) = {((2^nd ‘𝐵)‘𝑦), ((2^nd ‘𝐵)‘(𝑦 + 1))} ∧ ((iEdg‘𝐺)‘((1^st ‘𝐴)‘𝑦)) = {((2^nd ‘𝐴)‘𝑦), ((2^nd ‘𝐴)‘(𝑦 + 1))}) → ({((2^nd ‘𝐴)‘𝑦), ((2^nd ‘𝐴)‘(𝑦 + 1))} = {((2^nd ‘𝐵)‘𝑦), ((2^nd ‘𝐵)‘(𝑦 + 1))} → ((iEdg‘𝐺)‘((1^st ‘𝐵)‘𝑦)) = ((iEdg‘𝐺)‘((1^st ‘𝐴)‘𝑦))))
61	60	ral2imi 2609	. . . . . . . . . . . . . . . . 17 ⊢ (∀𝑦 ∈ (0..^𝑁)(((iEdg‘𝐺)‘((1^st ‘𝐵)‘𝑦)) = {((2^nd ‘𝐵)‘𝑦), ((2^nd ‘𝐵)‘(𝑦 + 1))} ∧ ((iEdg‘𝐺)‘((1^st ‘𝐴)‘𝑦)) = {((2^nd ‘𝐴)‘𝑦), ((2^nd ‘𝐴)‘(𝑦 + 1))}) → (∀𝑦 ∈ (0..^𝑁){((2^nd ‘𝐴)‘𝑦), ((2^nd ‘𝐴)‘(𝑦 + 1))} = {((2^nd ‘𝐵)‘𝑦), ((2^nd ‘𝐵)‘(𝑦 + 1))} → ∀𝑦 ∈ (0..^𝑁)((iEdg‘𝐺)‘((1^st ‘𝐵)‘𝑦)) = ((iEdg‘𝐺)‘((1^st ‘𝐴)‘𝑦))))
62	53, 61	sylbir 135	. . . . . . . . . . . . . . . 16 ⊢ ((∀𝑦 ∈ (0..^𝑁)((iEdg‘𝐺)‘((1^st ‘𝐵)‘𝑦)) = {((2^nd ‘𝐵)‘𝑦), ((2^nd ‘𝐵)‘(𝑦 + 1))} ∧ ∀𝑦 ∈ (0..^𝑁)((iEdg‘𝐺)‘((1^st ‘𝐴)‘𝑦)) = {((2^nd ‘𝐴)‘𝑦), ((2^nd ‘𝐴)‘(𝑦 + 1))}) → (∀𝑦 ∈ (0..^𝑁){((2^nd ‘𝐴)‘𝑦), ((2^nd ‘𝐴)‘(𝑦 + 1))} = {((2^nd ‘𝐵)‘𝑦), ((2^nd ‘𝐵)‘(𝑦 + 1))} → ∀𝑦 ∈ (0..^𝑁)((iEdg‘𝐺)‘((1^st ‘𝐵)‘𝑦)) = ((iEdg‘𝐺)‘((1^st ‘𝐴)‘𝑦))))
63	52, 62	biimtrdi 163	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 = (♯‘(1^st ‘𝐴)) ∧ 𝑁 = (♯‘(1^st ‘𝐵))) → ((∀𝑦 ∈ (0..^(♯‘(1^st ‘𝐵)))((iEdg‘𝐺)‘((1^st ‘𝐵)‘𝑦)) = {((2^nd ‘𝐵)‘𝑦), ((2^nd ‘𝐵)‘(𝑦 + 1))} ∧ ∀𝑦 ∈ (0..^(♯‘(1^st ‘𝐴)))((iEdg‘𝐺)‘((1^st ‘𝐴)‘𝑦)) = {((2^nd ‘𝐴)‘𝑦), ((2^nd ‘𝐴)‘(𝑦 + 1))}) → (∀𝑦 ∈ (0..^𝑁){((2^nd ‘𝐴)‘𝑦), ((2^nd ‘𝐴)‘(𝑦 + 1))} = {((2^nd ‘𝐵)‘𝑦), ((2^nd ‘𝐵)‘(𝑦 + 1))} → ∀𝑦 ∈ (0..^𝑁)((iEdg‘𝐺)‘((1^st ‘𝐵)‘𝑦)) = ((iEdg‘𝐺)‘((1^st ‘𝐴)‘𝑦)))))
64	63	com12 30	. . . . . . . . . . . . . 14 ⊢ ((∀𝑦 ∈ (0..^(♯‘(1^st ‘𝐵)))((iEdg‘𝐺)‘((1^st ‘𝐵)‘𝑦)) = {((2^nd ‘𝐵)‘𝑦), ((2^nd ‘𝐵)‘(𝑦 + 1))} ∧ ∀𝑦 ∈ (0..^(♯‘(1^st ‘𝐴)))((iEdg‘𝐺)‘((1^st ‘𝐴)‘𝑦)) = {((2^nd ‘𝐴)‘𝑦), ((2^nd ‘𝐴)‘(𝑦 + 1))}) → ((𝑁 = (♯‘(1^st ‘𝐴)) ∧ 𝑁 = (♯‘(1^st ‘𝐵))) → (∀𝑦 ∈ (0..^𝑁){((2^nd ‘𝐴)‘𝑦), ((2^nd ‘𝐴)‘(𝑦 + 1))} = {((2^nd ‘𝐵)‘𝑦), ((2^nd ‘𝐵)‘(𝑦 + 1))} → ∀𝑦 ∈ (0..^𝑁)((iEdg‘𝐺)‘((1^st ‘𝐵)‘𝑦)) = ((iEdg‘𝐺)‘((1^st ‘𝐴)‘𝑦)))))
65	64	ex 115	. . . . . . . . . . . . 13 ⊢ (∀𝑦 ∈ (0..^(♯‘(1^st ‘𝐵)))((iEdg‘𝐺)‘((1^st ‘𝐵)‘𝑦)) = {((2^nd ‘𝐵)‘𝑦), ((2^nd ‘𝐵)‘(𝑦 + 1))} → (∀𝑦 ∈ (0..^(♯‘(1^st ‘𝐴)))((iEdg‘𝐺)‘((1^st ‘𝐴)‘𝑦)) = {((2^nd ‘𝐴)‘𝑦), ((2^nd ‘𝐴)‘(𝑦 + 1))} → ((𝑁 = (♯‘(1^st ‘𝐴)) ∧ 𝑁 = (♯‘(1^st ‘𝐵))) → (∀𝑦 ∈ (0..^𝑁){((2^nd ‘𝐴)‘𝑦), ((2^nd ‘𝐴)‘(𝑦 + 1))} = {((2^nd ‘𝐵)‘𝑦), ((2^nd ‘𝐵)‘(𝑦 + 1))} → ∀𝑦 ∈ (0..^𝑁)((iEdg‘𝐺)‘((1^st ‘𝐵)‘𝑦)) = ((iEdg‘𝐺)‘((1^st ‘𝐴)‘𝑦))))))
66	65	3ad2ant3 1047	. . . . . . . . . . . 12 ⊢ (((1^st ‘𝐵) ∈ Word dom (iEdg‘𝐺) ∧ (2^nd ‘𝐵):(0...(♯‘(1^st ‘𝐵)))⟶(Vtx‘𝐺) ∧ ∀𝑦 ∈ (0..^(♯‘(1^st ‘𝐵)))((iEdg‘𝐺)‘((1^st ‘𝐵)‘𝑦)) = {((2^nd ‘𝐵)‘𝑦), ((2^nd ‘𝐵)‘(𝑦 + 1))}) → (∀𝑦 ∈ (0..^(♯‘(1^st ‘𝐴)))((iEdg‘𝐺)‘((1^st ‘𝐴)‘𝑦)) = {((2^nd ‘𝐴)‘𝑦), ((2^nd ‘𝐴)‘(𝑦 + 1))} → ((𝑁 = (♯‘(1^st ‘𝐴)) ∧ 𝑁 = (♯‘(1^st ‘𝐵))) → (∀𝑦 ∈ (0..^𝑁){((2^nd ‘𝐴)‘𝑦), ((2^nd ‘𝐴)‘(𝑦 + 1))} = {((2^nd ‘𝐵)‘𝑦), ((2^nd ‘𝐵)‘(𝑦 + 1))} → ∀𝑦 ∈ (0..^𝑁)((iEdg‘𝐺)‘((1^st ‘𝐵)‘𝑦)) = ((iEdg‘𝐺)‘((1^st ‘𝐴)‘𝑦))))))
67	66	com12 30	. . . . . . . . . . 11 ⊢ (∀𝑦 ∈ (0..^(♯‘(1^st ‘𝐴)))((iEdg‘𝐺)‘((1^st ‘𝐴)‘𝑦)) = {((2^nd ‘𝐴)‘𝑦), ((2^nd ‘𝐴)‘(𝑦 + 1))} → (((1^st ‘𝐵) ∈ Word dom (iEdg‘𝐺) ∧ (2^nd ‘𝐵):(0...(♯‘(1^st ‘𝐵)))⟶(Vtx‘𝐺) ∧ ∀𝑦 ∈ (0..^(♯‘(1^st ‘𝐵)))((iEdg‘𝐺)‘((1^st ‘𝐵)‘𝑦)) = {((2^nd ‘𝐵)‘𝑦), ((2^nd ‘𝐵)‘(𝑦 + 1))}) → ((𝑁 = (♯‘(1^st ‘𝐴)) ∧ 𝑁 = (♯‘(1^st ‘𝐵))) → (∀𝑦 ∈ (0..^𝑁){((2^nd ‘𝐴)‘𝑦), ((2^nd ‘𝐴)‘(𝑦 + 1))} = {((2^nd ‘𝐵)‘𝑦), ((2^nd ‘𝐵)‘(𝑦 + 1))} → ∀𝑦 ∈ (0..^𝑁)((iEdg‘𝐺)‘((1^st ‘𝐵)‘𝑦)) = ((iEdg‘𝐺)‘((1^st ‘𝐴)‘𝑦))))))
68	67	3ad2ant3 1047	. . . . . . . . . 10 ⊢ (((1^st ‘𝐴) ∈ Word dom (iEdg‘𝐺) ∧ (2^nd ‘𝐴):(0...(♯‘(1^st ‘𝐴)))⟶(Vtx‘𝐺) ∧ ∀𝑦 ∈ (0..^(♯‘(1^st ‘𝐴)))((iEdg‘𝐺)‘((1^st ‘𝐴)‘𝑦)) = {((2^nd ‘𝐴)‘𝑦), ((2^nd ‘𝐴)‘(𝑦 + 1))}) → (((1^st ‘𝐵) ∈ Word dom (iEdg‘𝐺) ∧ (2^nd ‘𝐵):(0...(♯‘(1^st ‘𝐵)))⟶(Vtx‘𝐺) ∧ ∀𝑦 ∈ (0..^(♯‘(1^st ‘𝐵)))((iEdg‘𝐺)‘((1^st ‘𝐵)‘𝑦)) = {((2^nd ‘𝐵)‘𝑦), ((2^nd ‘𝐵)‘(𝑦 + 1))}) → ((𝑁 = (♯‘(1^st ‘𝐴)) ∧ 𝑁 = (♯‘(1^st ‘𝐵))) → (∀𝑦 ∈ (0..^𝑁){((2^nd ‘𝐴)‘𝑦), ((2^nd ‘𝐴)‘(𝑦 + 1))} = {((2^nd ‘𝐵)‘𝑦), ((2^nd ‘𝐵)‘(𝑦 + 1))} → ∀𝑦 ∈ (0..^𝑁)((iEdg‘𝐺)‘((1^st ‘𝐵)‘𝑦)) = ((iEdg‘𝐺)‘((1^st ‘𝐴)‘𝑦))))))
69	68	imp 124	. . . . . . . . 9 ⊢ ((((1^st ‘𝐴) ∈ Word dom (iEdg‘𝐺) ∧ (2^nd ‘𝐴):(0...(♯‘(1^st ‘𝐴)))⟶(Vtx‘𝐺) ∧ ∀𝑦 ∈ (0..^(♯‘(1^st ‘𝐴)))((iEdg‘𝐺)‘((1^st ‘𝐴)‘𝑦)) = {((2^nd ‘𝐴)‘𝑦), ((2^nd ‘𝐴)‘(𝑦 + 1))}) ∧ ((1^st ‘𝐵) ∈ Word dom (iEdg‘𝐺) ∧ (2^nd ‘𝐵):(0...(♯‘(1^st ‘𝐵)))⟶(Vtx‘𝐺) ∧ ∀𝑦 ∈ (0..^(♯‘(1^st ‘𝐵)))((iEdg‘𝐺)‘((1^st ‘𝐵)‘𝑦)) = {((2^nd ‘𝐵)‘𝑦), ((2^nd ‘𝐵)‘(𝑦 + 1))})) → ((𝑁 = (♯‘(1^st ‘𝐴)) ∧ 𝑁 = (♯‘(1^st ‘𝐵))) → (∀𝑦 ∈ (0..^𝑁){((2^nd ‘𝐴)‘𝑦), ((2^nd ‘𝐴)‘(𝑦 + 1))} = {((2^nd ‘𝐵)‘𝑦), ((2^nd ‘𝐵)‘(𝑦 + 1))} → ∀𝑦 ∈ (0..^𝑁)((iEdg‘𝐺)‘((1^st ‘𝐵)‘𝑦)) = ((iEdg‘𝐺)‘((1^st ‘𝐴)‘𝑦)))))
70	69	expd 258	. . . . . . . 8 ⊢ ((((1^st ‘𝐴) ∈ Word dom (iEdg‘𝐺) ∧ (2^nd ‘𝐴):(0...(♯‘(1^st ‘𝐴)))⟶(Vtx‘𝐺) ∧ ∀𝑦 ∈ (0..^(♯‘(1^st ‘𝐴)))((iEdg‘𝐺)‘((1^st ‘𝐴)‘𝑦)) = {((2^nd ‘𝐴)‘𝑦), ((2^nd ‘𝐴)‘(𝑦 + 1))}) ∧ ((1^st ‘𝐵) ∈ Word dom (iEdg‘𝐺) ∧ (2^nd ‘𝐵):(0...(♯‘(1^st ‘𝐵)))⟶(Vtx‘𝐺) ∧ ∀𝑦 ∈ (0..^(♯‘(1^st ‘𝐵)))((iEdg‘𝐺)‘((1^st ‘𝐵)‘𝑦)) = {((2^nd ‘𝐵)‘𝑦), ((2^nd ‘𝐵)‘(𝑦 + 1))})) → (𝑁 = (♯‘(1^st ‘𝐴)) → (𝑁 = (♯‘(1^st ‘𝐵)) → (∀𝑦 ∈ (0..^𝑁){((2^nd ‘𝐴)‘𝑦), ((2^nd ‘𝐴)‘(𝑦 + 1))} = {((2^nd ‘𝐵)‘𝑦), ((2^nd ‘𝐵)‘(𝑦 + 1))} → ∀𝑦 ∈ (0..^𝑁)((iEdg‘𝐺)‘((1^st ‘𝐵)‘𝑦)) = ((iEdg‘𝐺)‘((1^st ‘𝐴)‘𝑦))))))
71	70	a1i 9	. . . . . . 7 ⊢ (𝐺 ∈ USPGraph → ((((1^st ‘𝐴) ∈ Word dom (iEdg‘𝐺) ∧ (2^nd ‘𝐴):(0...(♯‘(1^st ‘𝐴)))⟶(Vtx‘𝐺) ∧ ∀𝑦 ∈ (0..^(♯‘(1^st ‘𝐴)))((iEdg‘𝐺)‘((1^st ‘𝐴)‘𝑦)) = {((2^nd ‘𝐴)‘𝑦), ((2^nd ‘𝐴)‘(𝑦 + 1))}) ∧ ((1^st ‘𝐵) ∈ Word dom (iEdg‘𝐺) ∧ (2^nd ‘𝐵):(0...(♯‘(1^st ‘𝐵)))⟶(Vtx‘𝐺) ∧ ∀𝑦 ∈ (0..^(♯‘(1^st ‘𝐵)))((iEdg‘𝐺)‘((1^st ‘𝐵)‘𝑦)) = {((2^nd ‘𝐵)‘𝑦), ((2^nd ‘𝐵)‘(𝑦 + 1))})) → (𝑁 = (♯‘(1^st ‘𝐴)) → (𝑁 = (♯‘(1^st ‘𝐵)) → (∀𝑦 ∈ (0..^𝑁){((2^nd ‘𝐴)‘𝑦), ((2^nd ‘𝐴)‘(𝑦 + 1))} = {((2^nd ‘𝐵)‘𝑦), ((2^nd ‘𝐵)‘(𝑦 + 1))} → ∀𝑦 ∈ (0..^𝑁)((iEdg‘𝐺)‘((1^st ‘𝐵)‘𝑦)) = ((iEdg‘𝐺)‘((1^st ‘𝐴)‘𝑦)))))))
72	40, 45, 71	syl2and 295	. . . . . 6 ⊢ (𝐺 ∈ USPGraph → ((𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) → (𝑁 = (♯‘(1^st ‘𝐴)) → (𝑁 = (♯‘(1^st ‘𝐵)) → (∀𝑦 ∈ (0..^𝑁){((2^nd ‘𝐴)‘𝑦), ((2^nd ‘𝐴)‘(𝑦 + 1))} = {((2^nd ‘𝐵)‘𝑦), ((2^nd ‘𝐵)‘(𝑦 + 1))} → ∀𝑦 ∈ (0..^𝑁)((iEdg‘𝐺)‘((1^st ‘𝐵)‘𝑦)) = ((iEdg‘𝐺)‘((1^st ‘𝐴)‘𝑦)))))))
73	72	3imp1 1247	. . . . 5 ⊢ (((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 𝑁 = (♯‘(1^st ‘𝐴))) ∧ 𝑁 = (♯‘(1^st ‘𝐵))) → (∀𝑦 ∈ (0..^𝑁){((2^nd ‘𝐴)‘𝑦), ((2^nd ‘𝐴)‘(𝑦 + 1))} = {((2^nd ‘𝐵)‘𝑦), ((2^nd ‘𝐵)‘(𝑦 + 1))} → ∀𝑦 ∈ (0..^𝑁)((iEdg‘𝐺)‘((1^st ‘𝐵)‘𝑦)) = ((iEdg‘𝐺)‘((1^st ‘𝐴)‘𝑦))))
74		eqcom 2236	. . . . . . 7 ⊢ (((iEdg‘𝐺)‘((1^st ‘𝐵)‘𝑦)) = ((iEdg‘𝐺)‘((1^st ‘𝐴)‘𝑦)) ↔ ((iEdg‘𝐺)‘((1^st ‘𝐴)‘𝑦)) = ((iEdg‘𝐺)‘((1^st ‘𝐵)‘𝑦)))
75	35	uspgrf1oedg 16297	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ USPGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1-onto→(Edg‘𝐺))
76		f1of1 5618	. . . . . . . . . . . 12 ⊢ ((iEdg‘𝐺):dom (iEdg‘𝐺)–1-1-onto→(Edg‘𝐺) → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→(Edg‘𝐺))
77	75, 76	syl 14	. . . . . . . . . . 11 ⊢ (𝐺 ∈ USPGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→(Edg‘𝐺))
78		eqidd 2235	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ USPGraph → (iEdg‘𝐺) = (iEdg‘𝐺))
79		eqidd 2235	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ USPGraph → dom (iEdg‘𝐺) = dom (iEdg‘𝐺))
80		edgvalg 16180	. . . . . . . . . . . . 13 ⊢ (𝐺 ∈ USPGraph → (Edg‘𝐺) = ran (iEdg‘𝐺))
81	80	eqcomd 2240	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ USPGraph → ran (iEdg‘𝐺) = (Edg‘𝐺))
82	78, 79, 81	f1eq123d 5611	. . . . . . . . . . 11 ⊢ (𝐺 ∈ USPGraph → ((iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→ran (iEdg‘𝐺) ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→(Edg‘𝐺)))
83	77, 82	mpbird 167	. . . . . . . . . 10 ⊢ (𝐺 ∈ USPGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→ran (iEdg‘𝐺))
84	83	3ad2ant1 1045	. . . . . . . . 9 ⊢ ((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 𝑁 = (♯‘(1^st ‘𝐴))) → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→ran (iEdg‘𝐺))
85	84	adantr 276	. . . . . . . 8 ⊢ (((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 𝑁 = (♯‘(1^st ‘𝐴))) ∧ 𝑁 = (♯‘(1^st ‘𝐵))) → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→ran (iEdg‘𝐺))
86	34, 35, 36, 37	wlkelwrd 16474	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ (Walks‘𝐺) → ((1^st ‘𝐴) ∈ Word dom (iEdg‘𝐺) ∧ (2^nd ‘𝐴):(0...(♯‘(1^st ‘𝐴)))⟶(Vtx‘𝐺)))
87	34, 35, 41, 42	wlkelwrd 16474	. . . . . . . . . . . . . . 15 ⊢ (𝐵 ∈ (Walks‘𝐺) → ((1^st ‘𝐵) ∈ Word dom (iEdg‘𝐺) ∧ (2^nd ‘𝐵):(0...(♯‘(1^st ‘𝐵)))⟶(Vtx‘𝐺)))
88		oveq2 6066	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑁 = (♯‘(1^st ‘𝐴)) → (0..^𝑁) = (0..^(♯‘(1^st ‘𝐴))))
89	88	eleq2d 2304	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑁 = (♯‘(1^st ‘𝐴)) → (𝑦 ∈ (0..^𝑁) ↔ 𝑦 ∈ (0..^(♯‘(1^st ‘𝐴)))))
90		wrdsymbcl 11263	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((1^st ‘𝐴) ∈ Word dom (iEdg‘𝐺) ∧ 𝑦 ∈ (0..^(♯‘(1^st ‘𝐴)))) → ((1^st ‘𝐴)‘𝑦) ∈ dom (iEdg‘𝐺))
91	90	expcom 116	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑦 ∈ (0..^(♯‘(1^st ‘𝐴))) → ((1^st ‘𝐴) ∈ Word dom (iEdg‘𝐺) → ((1^st ‘𝐴)‘𝑦) ∈ dom (iEdg‘𝐺)))
92	89, 91	biimtrdi 163	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑁 = (♯‘(1^st ‘𝐴)) → (𝑦 ∈ (0..^𝑁) → ((1^st ‘𝐴) ∈ Word dom (iEdg‘𝐺) → ((1^st ‘𝐴)‘𝑦) ∈ dom (iEdg‘𝐺))))
93	92	adantr 276	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑁 = (♯‘(1^st ‘𝐴)) ∧ 𝑁 = (♯‘(1^st ‘𝐵))) → (𝑦 ∈ (0..^𝑁) → ((1^st ‘𝐴) ∈ Word dom (iEdg‘𝐺) → ((1^st ‘𝐴)‘𝑦) ∈ dom (iEdg‘𝐺))))
94	93	imp 124	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑁 = (♯‘(1^st ‘𝐴)) ∧ 𝑁 = (♯‘(1^st ‘𝐵))) ∧ 𝑦 ∈ (0..^𝑁)) → ((1^st ‘𝐴) ∈ Word dom (iEdg‘𝐺) → ((1^st ‘𝐴)‘𝑦) ∈ dom (iEdg‘𝐺)))
95	94	com12 30	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((1^st ‘𝐴) ∈ Word dom (iEdg‘𝐺) → (((𝑁 = (♯‘(1^st ‘𝐴)) ∧ 𝑁 = (♯‘(1^st ‘𝐵))) ∧ 𝑦 ∈ (0..^𝑁)) → ((1^st ‘𝐴)‘𝑦) ∈ dom (iEdg‘𝐺)))
96	95	adantl 277	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((1^st ‘𝐵) ∈ Word dom (iEdg‘𝐺) ∧ (1^st ‘𝐴) ∈ Word dom (iEdg‘𝐺)) → (((𝑁 = (♯‘(1^st ‘𝐴)) ∧ 𝑁 = (♯‘(1^st ‘𝐵))) ∧ 𝑦 ∈ (0..^𝑁)) → ((1^st ‘𝐴)‘𝑦) ∈ dom (iEdg‘𝐺)))
97		oveq2 6066	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑁 = (♯‘(1^st ‘𝐵)) → (0..^𝑁) = (0..^(♯‘(1^st ‘𝐵))))
98	97	eleq2d 2304	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑁 = (♯‘(1^st ‘𝐵)) → (𝑦 ∈ (0..^𝑁) ↔ 𝑦 ∈ (0..^(♯‘(1^st ‘𝐵)))))
99		wrdsymbcl 11263	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((1^st ‘𝐵) ∈ Word dom (iEdg‘𝐺) ∧ 𝑦 ∈ (0..^(♯‘(1^st ‘𝐵)))) → ((1^st ‘𝐵)‘𝑦) ∈ dom (iEdg‘𝐺))
100	99	expcom 116	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑦 ∈ (0..^(♯‘(1^st ‘𝐵))) → ((1^st ‘𝐵) ∈ Word dom (iEdg‘𝐺) → ((1^st ‘𝐵)‘𝑦) ∈ dom (iEdg‘𝐺)))
101	98, 100	biimtrdi 163	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑁 = (♯‘(1^st ‘𝐵)) → (𝑦 ∈ (0..^𝑁) → ((1^st ‘𝐵) ∈ Word dom (iEdg‘𝐺) → ((1^st ‘𝐵)‘𝑦) ∈ dom (iEdg‘𝐺))))
102	101	adantl 277	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑁 = (♯‘(1^st ‘𝐴)) ∧ 𝑁 = (♯‘(1^st ‘𝐵))) → (𝑦 ∈ (0..^𝑁) → ((1^st ‘𝐵) ∈ Word dom (iEdg‘𝐺) → ((1^st ‘𝐵)‘𝑦) ∈ dom (iEdg‘𝐺))))
103	102	imp 124	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑁 = (♯‘(1^st ‘𝐴)) ∧ 𝑁 = (♯‘(1^st ‘𝐵))) ∧ 𝑦 ∈ (0..^𝑁)) → ((1^st ‘𝐵) ∈ Word dom (iEdg‘𝐺) → ((1^st ‘𝐵)‘𝑦) ∈ dom (iEdg‘𝐺)))
104	103	com12 30	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((1^st ‘𝐵) ∈ Word dom (iEdg‘𝐺) → (((𝑁 = (♯‘(1^st ‘𝐴)) ∧ 𝑁 = (♯‘(1^st ‘𝐵))) ∧ 𝑦 ∈ (0..^𝑁)) → ((1^st ‘𝐵)‘𝑦) ∈ dom (iEdg‘𝐺)))
105	104	adantr 276	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((1^st ‘𝐵) ∈ Word dom (iEdg‘𝐺) ∧ (1^st ‘𝐴) ∈ Word dom (iEdg‘𝐺)) → (((𝑁 = (♯‘(1^st ‘𝐴)) ∧ 𝑁 = (♯‘(1^st ‘𝐵))) ∧ 𝑦 ∈ (0..^𝑁)) → ((1^st ‘𝐵)‘𝑦) ∈ dom (iEdg‘𝐺)))
106	96, 105	jcad 307	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((1^st ‘𝐵) ∈ Word dom (iEdg‘𝐺) ∧ (1^st ‘𝐴) ∈ Word dom (iEdg‘𝐺)) → (((𝑁 = (♯‘(1^st ‘𝐴)) ∧ 𝑁 = (♯‘(1^st ‘𝐵))) ∧ 𝑦 ∈ (0..^𝑁)) → (((1^st ‘𝐴)‘𝑦) ∈ dom (iEdg‘𝐺) ∧ ((1^st ‘𝐵)‘𝑦) ∈ dom (iEdg‘𝐺))))
107	106	ex 115	. . . . . . . . . . . . . . . . . . 19 ⊢ ((1^st ‘𝐵) ∈ Word dom (iEdg‘𝐺) → ((1^st ‘𝐴) ∈ Word dom (iEdg‘𝐺) → (((𝑁 = (♯‘(1^st ‘𝐴)) ∧ 𝑁 = (♯‘(1^st ‘𝐵))) ∧ 𝑦 ∈ (0..^𝑁)) → (((1^st ‘𝐴)‘𝑦) ∈ dom (iEdg‘𝐺) ∧ ((1^st ‘𝐵)‘𝑦) ∈ dom (iEdg‘𝐺)))))
108	107	adantr 276	. . . . . . . . . . . . . . . . . 18 ⊢ (((1^st ‘𝐵) ∈ Word dom (iEdg‘𝐺) ∧ (2^nd ‘𝐵):(0...(♯‘(1^st ‘𝐵)))⟶(Vtx‘𝐺)) → ((1^st ‘𝐴) ∈ Word dom (iEdg‘𝐺) → (((𝑁 = (♯‘(1^st ‘𝐴)) ∧ 𝑁 = (♯‘(1^st ‘𝐵))) ∧ 𝑦 ∈ (0..^𝑁)) → (((1^st ‘𝐴)‘𝑦) ∈ dom (iEdg‘𝐺) ∧ ((1^st ‘𝐵)‘𝑦) ∈ dom (iEdg‘𝐺)))))
109	108	com12 30	. . . . . . . . . . . . . . . . 17 ⊢ ((1^st ‘𝐴) ∈ Word dom (iEdg‘𝐺) → (((1^st ‘𝐵) ∈ Word dom (iEdg‘𝐺) ∧ (2^nd ‘𝐵):(0...(♯‘(1^st ‘𝐵)))⟶(Vtx‘𝐺)) → (((𝑁 = (♯‘(1^st ‘𝐴)) ∧ 𝑁 = (♯‘(1^st ‘𝐵))) ∧ 𝑦 ∈ (0..^𝑁)) → (((1^st ‘𝐴)‘𝑦) ∈ dom (iEdg‘𝐺) ∧ ((1^st ‘𝐵)‘𝑦) ∈ dom (iEdg‘𝐺)))))
110	109	adantr 276	. . . . . . . . . . . . . . . 16 ⊢ (((1^st ‘𝐴) ∈ Word dom (iEdg‘𝐺) ∧ (2^nd ‘𝐴):(0...(♯‘(1^st ‘𝐴)))⟶(Vtx‘𝐺)) → (((1^st ‘𝐵) ∈ Word dom (iEdg‘𝐺) ∧ (2^nd ‘𝐵):(0...(♯‘(1^st ‘𝐵)))⟶(Vtx‘𝐺)) → (((𝑁 = (♯‘(1^st ‘𝐴)) ∧ 𝑁 = (♯‘(1^st ‘𝐵))) ∧ 𝑦 ∈ (0..^𝑁)) → (((1^st ‘𝐴)‘𝑦) ∈ dom (iEdg‘𝐺) ∧ ((1^st ‘𝐵)‘𝑦) ∈ dom (iEdg‘𝐺)))))
111	110	imp 124	. . . . . . . . . . . . . . 15 ⊢ ((((1^st ‘𝐴) ∈ Word dom (iEdg‘𝐺) ∧ (2^nd ‘𝐴):(0...(♯‘(1^st ‘𝐴)))⟶(Vtx‘𝐺)) ∧ ((1^st ‘𝐵) ∈ Word dom (iEdg‘𝐺) ∧ (2^nd ‘𝐵):(0...(♯‘(1^st ‘𝐵)))⟶(Vtx‘𝐺))) → (((𝑁 = (♯‘(1^st ‘𝐴)) ∧ 𝑁 = (♯‘(1^st ‘𝐵))) ∧ 𝑦 ∈ (0..^𝑁)) → (((1^st ‘𝐴)‘𝑦) ∈ dom (iEdg‘𝐺) ∧ ((1^st ‘𝐵)‘𝑦) ∈ dom (iEdg‘𝐺))))
112	86, 87, 111	syl2an 289	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) → (((𝑁 = (♯‘(1^st ‘𝐴)) ∧ 𝑁 = (♯‘(1^st ‘𝐵))) ∧ 𝑦 ∈ (0..^𝑁)) → (((1^st ‘𝐴)‘𝑦) ∈ dom (iEdg‘𝐺) ∧ ((1^st ‘𝐵)‘𝑦) ∈ dom (iEdg‘𝐺))))
113	112	expd 258	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) → ((𝑁 = (♯‘(1^st ‘𝐴)) ∧ 𝑁 = (♯‘(1^st ‘𝐵))) → (𝑦 ∈ (0..^𝑁) → (((1^st ‘𝐴)‘𝑦) ∈ dom (iEdg‘𝐺) ∧ ((1^st ‘𝐵)‘𝑦) ∈ dom (iEdg‘𝐺)))))
114	113	expd 258	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) → (𝑁 = (♯‘(1^st ‘𝐴)) → (𝑁 = (♯‘(1^st ‘𝐵)) → (𝑦 ∈ (0..^𝑁) → (((1^st ‘𝐴)‘𝑦) ∈ dom (iEdg‘𝐺) ∧ ((1^st ‘𝐵)‘𝑦) ∈ dom (iEdg‘𝐺))))))
115	114	imp 124	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 𝑁 = (♯‘(1^st ‘𝐴))) → (𝑁 = (♯‘(1^st ‘𝐵)) → (𝑦 ∈ (0..^𝑁) → (((1^st ‘𝐴)‘𝑦) ∈ dom (iEdg‘𝐺) ∧ ((1^st ‘𝐵)‘𝑦) ∈ dom (iEdg‘𝐺)))))
116	115	3adant1 1042	. . . . . . . . . 10 ⊢ ((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 𝑁 = (♯‘(1^st ‘𝐴))) → (𝑁 = (♯‘(1^st ‘𝐵)) → (𝑦 ∈ (0..^𝑁) → (((1^st ‘𝐴)‘𝑦) ∈ dom (iEdg‘𝐺) ∧ ((1^st ‘𝐵)‘𝑦) ∈ dom (iEdg‘𝐺)))))
117	116	imp 124	. . . . . . . . 9 ⊢ (((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 𝑁 = (♯‘(1^st ‘𝐴))) ∧ 𝑁 = (♯‘(1^st ‘𝐵))) → (𝑦 ∈ (0..^𝑁) → (((1^st ‘𝐴)‘𝑦) ∈ dom (iEdg‘𝐺) ∧ ((1^st ‘𝐵)‘𝑦) ∈ dom (iEdg‘𝐺))))
118	117	imp 124	. . . . . . . 8 ⊢ ((((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 𝑁 = (♯‘(1^st ‘𝐴))) ∧ 𝑁 = (♯‘(1^st ‘𝐵))) ∧ 𝑦 ∈ (0..^𝑁)) → (((1^st ‘𝐴)‘𝑦) ∈ dom (iEdg‘𝐺) ∧ ((1^st ‘𝐵)‘𝑦) ∈ dom (iEdg‘𝐺)))
119		f1veqaeq 5948	. . . . . . . 8 ⊢ (((iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→ran (iEdg‘𝐺) ∧ (((1^st ‘𝐴)‘𝑦) ∈ dom (iEdg‘𝐺) ∧ ((1^st ‘𝐵)‘𝑦) ∈ dom (iEdg‘𝐺))) → (((iEdg‘𝐺)‘((1^st ‘𝐴)‘𝑦)) = ((iEdg‘𝐺)‘((1^st ‘𝐵)‘𝑦)) → ((1^st ‘𝐴)‘𝑦) = ((1^st ‘𝐵)‘𝑦)))
120	85, 118, 119	syl2an2r 599	. . . . . . 7 ⊢ ((((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 𝑁 = (♯‘(1^st ‘𝐴))) ∧ 𝑁 = (♯‘(1^st ‘𝐵))) ∧ 𝑦 ∈ (0..^𝑁)) → (((iEdg‘𝐺)‘((1^st ‘𝐴)‘𝑦)) = ((iEdg‘𝐺)‘((1^st ‘𝐵)‘𝑦)) → ((1^st ‘𝐴)‘𝑦) = ((1^st ‘𝐵)‘𝑦)))
121	74, 120	biimtrid 152	. . . . . 6 ⊢ ((((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 𝑁 = (♯‘(1^st ‘𝐴))) ∧ 𝑁 = (♯‘(1^st ‘𝐵))) ∧ 𝑦 ∈ (0..^𝑁)) → (((iEdg‘𝐺)‘((1^st ‘𝐵)‘𝑦)) = ((iEdg‘𝐺)‘((1^st ‘𝐴)‘𝑦)) → ((1^st ‘𝐴)‘𝑦) = ((1^st ‘𝐵)‘𝑦)))
122	121	ralimdva 2611	. . . . 5 ⊢ (((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 𝑁 = (♯‘(1^st ‘𝐴))) ∧ 𝑁 = (♯‘(1^st ‘𝐵))) → (∀𝑦 ∈ (0..^𝑁)((iEdg‘𝐺)‘((1^st ‘𝐵)‘𝑦)) = ((iEdg‘𝐺)‘((1^st ‘𝐴)‘𝑦)) → ∀𝑦 ∈ (0..^𝑁)((1^st ‘𝐴)‘𝑦) = ((1^st ‘𝐵)‘𝑦)))
123	32, 73, 122	3syld 57	. . . 4 ⊢ (((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 𝑁 = (♯‘(1^st ‘𝐴))) ∧ 𝑁 = (♯‘(1^st ‘𝐵))) → (∀𝑦 ∈ (0...𝑁)((2^nd ‘𝐴)‘𝑦) = ((2^nd ‘𝐵)‘𝑦) → ∀𝑦 ∈ (0..^𝑁)((1^st ‘𝐴)‘𝑦) = ((1^st ‘𝐵)‘𝑦)))
124	123	expimpd 363	. . 3 ⊢ ((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 𝑁 = (♯‘(1^st ‘𝐴))) → ((𝑁 = (♯‘(1^st ‘𝐵)) ∧ ∀𝑦 ∈ (0...𝑁)((2^nd ‘𝐴)‘𝑦) = ((2^nd ‘𝐵)‘𝑦)) → ∀𝑦 ∈ (0..^𝑁)((1^st ‘𝐴)‘𝑦) = ((1^st ‘𝐵)‘𝑦)))
125	124	pm4.71d 393	. 2 ⊢ ((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 𝑁 = (♯‘(1^st ‘𝐴))) → ((𝑁 = (♯‘(1^st ‘𝐵)) ∧ ∀𝑦 ∈ (0...𝑁)((2^nd ‘𝐴)‘𝑦) = ((2^nd ‘𝐵)‘𝑦)) ↔ ((𝑁 = (♯‘(1^st ‘𝐵)) ∧ ∀𝑦 ∈ (0...𝑁)((2^nd ‘𝐴)‘𝑦) = ((2^nd ‘𝐵)‘𝑦)) ∧ ∀𝑦 ∈ (0..^𝑁)((1^st ‘𝐴)‘𝑦) = ((1^st ‘𝐵)‘𝑦))))
126	2, 5, 125	3bitr4d 220	1 ⊢ ((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 𝑁 = (♯‘(1^st ‘𝐴))) → (𝐴 = 𝐵 ↔ (𝑁 = (♯‘(1^st ‘𝐵)) ∧ ∀𝑦 ∈ (0...𝑁)((2^nd ‘𝐴)‘𝑦) = ((2^nd ‘𝐵)‘𝑦))))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∧ w3a 1005 = wceq 1398 ∈ wcel 2205 ∀wral 2522 ⊆ wss 3214 {cpr 3695 dom cdm 4754 ran crn 4755 ⟶wf 5353 –1-1→wf1 5354 –1-1-onto→wf1o 5356 ‘cfv 5357 (class class class)co 6058 1^st c1st 6345 2^nd c2nd 6346 0cc0 8143 1c1 8144 + caddc 8146 ...cfz 10361 ..^cfzo 10498 ♯chash 11163 Word cword 11249 Vtxcvtx 16133 iEdgciedg 16134 Edgcedg 16178 UPGraphcupgr 16212 USPGraphcuspgr 16274 Walkscwlks 16438

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2207 ax-14 2208 ax-ext 2216 ax-coll 4230 ax-sep 4233 ax-nul 4241 ax-pow 4292 ax-pr 4327 ax-un 4559 ax-setind 4664 ax-iinf 4715 ax-cnex 8234 ax-resscn 8235 ax-1cn 8236 ax-1re 8237 ax-icn 8238 ax-addcl 8239 ax-addrcl 8240 ax-mulcl 8241 ax-addcom 8243 ax-mulcom 8244 ax-addass 8245 ax-mulass 8246 ax-distr 8247 ax-i2m1 8248 ax-0lt1 8249 ax-1rid 8250 ax-0id 8251 ax-rnegex 8252 ax-cnre 8254 ax-pre-ltirr 8255 ax-pre-ltwlin 8256 ax-pre-lttrn 8257 ax-pre-apti 8258 ax-pre-ltadd 8259

This theorem depends on definitions: df-bi 117 df-dc 843 df-ifp 987 df-3or 1006 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1812 df-eu 2085 df-mo 2086 df-clab 2221 df-cleq 2227 df-clel 2230 df-nfc 2375 df-ne 2415 df-nel 2510 df-ral 2527 df-rex 2528 df-reu 2529 df-rab 2531 df-v 2817 df-sbc 3046 df-csb 3142 df-dif 3216 df-un 3218 df-in 3220 df-ss 3227 df-nul 3513 df-if 3625 df-pw 3676 df-sn 3700 df-pr 3701 df-op 3703 df-uni 3920 df-int 3955 df-iun 3998 df-br 4115 df-opab 4177 df-mpt 4178 df-tr 4214 df-id 4419 df-iord 4492 df-on 4494 df-ilim 4495 df-suc 4497 df-iom 4718 df-xp 4760 df-rel 4761 df-cnv 4762 df-co 4763 df-dm 4764 df-rn 4765 df-res 4766 df-ima 4767 df-iota 5317 df-fun 5359 df-fn 5360 df-f 5361 df-f1 5362 df-fo 5363 df-f1o 5364 df-fv 5365 df-riota 6011 df-ov 6061 df-oprab 6062 df-mpo 6063 df-1st 6347 df-2nd 6348 df-recs 6549 df-frec 6635 df-1o 6660 df-2o 6661 df-er 6780 df-map 6897 df-en 6989 df-dom 6990 df-fin 6991 df-pnf 8326 df-mnf 8327 df-xr 8328 df-ltxr 8329 df-le 8330 df-sub 8462 df-neg 8463 df-inn 9255 df-2 9313 df-3 9314 df-4 9315 df-5 9316 df-6 9317 df-7 9318 df-8 9319 df-9 9320 df-n0 9514 df-z 9595 df-dec 9728 df-uz 9872 df-fz 10362 df-fzo 10499 df-ihash 11164 df-word 11250 df-ndx 13299 df-slot 13300 df-base 13302 df-edgf 16126 df-vtx 16135 df-iedg 16136 df-edg 16179 df-uhgrm 16190 df-upgren 16214 df-uspgren 16276 df-wlks 16439

This theorem is referenced by: uspgr2wlkeq2 16487

W3C validator